Linguistics models for system analysis- Chuluundorj.B
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Linguistics: models forsystem analysis Academician Chuluundorj. B President, University of the Humanities 1
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TYPES OF SYSTEMS Universal categories Space TimeMechanistic Energy System + Change Entropy Causation 2
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Types of System Parts Whole ExampleModelMechanistic No choice No choice MachinesAnimate No choice Choice PersonsSocial Choice Choice CorporationsEcological Choice No choice Mature 3
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TYPES OF SYSTEMClosed system is independent of its environmentSemi-closed system A thermostat, circulatory systemOpen system organization systems interact with the outside world exchanging information, energy or material 4
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Homeostatic Systems Or Dynamic Equilibrium: Types of System Example of System Homeostatic Human body Closed A closed economy of a country Semi-closed A circulatory system or heating system Open A business Deterministic A computerized accounting system Probabilistic A football playing system Cybernetic Most businesses 5
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SYSTEMS THEORY Systems theory is the interdisciplinary study of systems: - Bertalanffy‟s general system theory (GST) - Action theory of Talcott Parsons - Social systems of Niklas Luhmann Self-regulating systems: - Physiological system of our body - Human learning processesChaos Theory – The behavior of certain dynamical systems (as the butterfly effect) 6
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COMPLEX ADAPTIVE SYSTEMMacroscopic collection‟ of relatively „similar and partially connected micro-structures - to adapt to the changing environment - dynamic networks ofinteractions, equilibrium conditions 7
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DYNAMIC SYSTEMS THEORYDynamic systems theory is an area of mathemitics used to describe the behavior of complex dynamical systems, usually by employing differential equation(continuous dynamical systems) or difference equations(discrete dynamical systems) The Lorenz attractor is an example of a non-linear dynamical system. 8
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A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space.Small changes in the state of the system correspond to small changes in the numbers.The numbers are the coordinates of a geometrical space – a manifold.Symbolic dynamics - a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator 9
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Dynamical system theory – the neo-Piagetian theories of cognitive developmentThe learner‟s mind - A state of disequilibrium The spontaneous creation of coherent forms Newly formed macroscopic and microscopic structure support each other, speeding up the process. 10
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STRUCTUREStructure – system is made of configuration of items, a collection of inter- related components, network featuring many-to-many links. 11
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Structural systemA structural system one-dimensional, three-dimensional depending onthe space dimension.Real-world structure is strictly three-dimensional.Verbal thinking is oriented to description of real world structuremost of mental models have reflected real world structure – threedimensional structure. 12
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FUNCTION f(x) = … is the classic way of writing a function. And there are other ways, as you will see The Input The Relationship The Output Input Relationship Output 0 x2 0 1 x2 2 7 x2 14 10 x2 20 … … … 13
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Injective, surjective, and bijectiveA function is way of matching the members of a set “A” to a set “B”:Multiplicative function: preserves the multiplication operationContinuous function: in which preimages of open sets are open f(xy) = f(x)f(y)Composite function: and be two functions. The composition of fand g defines a function such that . 14
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SemanticsGeneral semantics (GS) - relations between the non-verbal and the verbal, including our verbal and nonverbal transactionsGS - time binding - human engineeringGS - the meeting point of scientific-mathematical methods and daily lifeThe basic unit of study for general semantics - Human evaluational (or semantic) reactionsEvaluational reactions neurologically based responses to words, symbols, and other eventsMeaning - ideas – mental representations - truth conditions (T/F – Semantics and Pragmatics) - semantic externalism (reference) - determined by the consequences of its application (pragmatic theory) - - prototype - radial structures - in relation to other concepts and mental states (conceptual role semantics)Semantic features. bachelor [+HUMAN, +MALE, +ADULT, +NEVER-MARRIED (?!)]. 15
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Semantics and PragmaticsThe infinite cardinality of emergent propositions in a like-quantum semantics.Hilbert space + Fuzzy theory – a set to a variable degree of membership; a proposition and its variable relation to the true and false logical constants. (New version of pragmatics T/F)Quantum coherence: . The fuzzy interpretation the properties and belong to with degrees of membership and respectively. It means for complex systems: the Schrodinger’s cat can be simultaneously both alive and dead! 16
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PragmaticsS - a finite set of signalsT - a finite set of types (information states) the sender might be in - a prior probability distribution over - a truth relation between and SA - a set of actions that receiver may take and - utility functions for sender and receiver to map triples from to real numbers 17
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Riemann’s sphere – a complex amplitude (Dirac, 1947) each point on the sphere fixes a single interpretation of a given situation ,i.e. the assigning of a coherent set of truth-values to a given proposition. Amplitude between the logical description of the two worlds – is expressed by (1+cos ), where is the angle between the two interpretations. (Semantics/Pragmatics) 18
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System analysis: Ideas and modelsContinuum Hypothesis (Cantor)The natural numbers are in the set The size of this set, its cardinality, isinfiniteThus /E/=N. The sets of real numbers - the continuum, “R”-the set of all realnumbers or the continuum.Ideas: Finite set be applied to measuring number of meanings Infinite set if words are finite set, meanings are infinite set.Comments: mental states (бодол) – infinite set of sentences presenting these states – infinite set of words Finite set of rules 19
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… Cantor’s theorem: The cardinality of the power set P(S) (set of all subsets of S) is greater than the cardinality of S. In symbols │S│< │ (S) │. Cantor’s theorem establishes a hierarchy of sets with infinite cardinalities: … Ideas: Notion of “set” be applied to classification of words. Notion of “continuum” (N as a proper subset of R) be applied to mental lexicon and thus to measuring capacity of semantic memory. Sum of propositions in the discourse (set) and sum of subsets of propositions in the discourse (set). 20
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Russell’s paradox:Comment: “X is Red” is equivalent to “X is a member of the set of red things”.The set B of bananas is not itself a banana.Idea: to be applied to the invariant theory (Boolean algebra, Cayley-Hamilton theorems , Hilbert spaces) to invariance in semantics.The characteristic polynomial of A is:The Cayley-Hamilton theorem states that “substituting” the matrix A for in this polynomial results in the zero matrix:Hilbert spaces-extends the methods of vector algebra and calculus from the two dimensional Euclidean plane and three dimensional space to space with any finite or infinite number of dimensions.Mental spaces-Semantic spaces-embedding Non-Euclidean space 21
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Comments:Version I. The intermediate value theorem Bolzano’s theorem states the following: If f is a real-valued continuous function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such that f(c) = u. The intermediate value theoremIdeas: Semantic space (mental space semantic space)words (or setof words) in semantic space.Invariance (invariant word) and variant words as elements of set (prototypeand radial structures) should be modeled in terms of continuous function(Bolzano’s theorem). 22
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Isomorphism: Two structures are isomorphic when:They have the same number of elements or objectsThe relations among elements of the one structure have same pattern asthe relations among the elements of the other.Homomorphism is a weaker notion-requires the second condition, but notthe first to have different number of elements.Isomorphism is structure preserving mapping.Homomorphism is a topological isomorphism. 23
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Embedding is one instance of mathematical structure contained within another instance such as group that is a subgroup.Idea: to apply to the analysis of vocabulary in combination with Russell’s paradox + Continuum hypothesis (Cantor) and Cantor’s theorem+ Bolzano’s theorem. ХСЭ ном p 11-19to reduce dual frame to a much smaller dimensional space through natural embeddings(reduction of large spaces to finite dimensional Hilbert spaces).Embedding: in algebra (structure preserving map in category theory- morphism) in topology (injective continuous map-“x” as a subspace of “y”- homomorphism)So: One space X is embedded in another space Y when the properties of Y restricted to X are the same as the properties of X in terms of typologically different languages.Cross-language embedding-to establishing an equivalence of structures and classes.Local isometry between Riemannian manifolds Riemannian symmetric spacesSymmetry/assymmetry of Linguistic structures. 24
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Differential geometry: distance between two events in space-time - its dependency on particular coordinate system general relativity.Intrinsic features characterize the surface independently of any particular coordinatization systems.Intrinsic features of space-time (curvature, metric tensor) are objectively real.Extrinsic features are mere artifacts of the form of representation of subjective coordinatization, particularly of verbal thinking spaces.Idea: Intrinsic features are objective, but in terms of interpretation by (subject) may have some influence. So these two factors have caused semantic changes, transformations, pragmatic interpretations.Relativity event structure is same for all.Mapping this event in the brain (verbal mapping) is differing, varying.to test: SOV, SVO- is a matter of extrinsic features (?).Extrinsic features – color recognition (?) Стол стoитIntrinsic features are identic, but coordinatization system is ? - Ковер лежитIdea: Sapir-Whorf hypothesis of linguistic relativity be renewed in terms of differential geometry + topology. 25
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A Structure is the abstract form of a system focusing on the interrelationships among objects and ignoring any features of them.Set theory axiom: two primitive concepts-set and member.A set is a collection of objects. But set’s identity is wholly dependent of its members-change the member and you change the set. By contrast, groups very nicely fit the structuralist account.Idea: Structure must be presented on the form of sets (set-based model for structural analysis) Group theory for structural analysis 26
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In projective geometry: every pair of lines intersect at a point, the exceptions are parallel lines with the introduction of a point of infinity, even parallel lines can intersect .Ideas: In the discourse a propositions (as a lines, vectors) have exactly one point in common and they intersect at this point. (Super proposition?).Discourse as a set of propositions (algebra).Propositions have their directions (links) and magnitude (intensity). So propositions should be modeled in terms of vectors.Implicitness of propositions Non-Euclidean geometry.be applied to a perception of music, to interpretation (verbal) of paintings. 27
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Formalism in mathematics: Knot theory provides different forms of representation .Knot-a closed, non-intersecting curve in space. Knot can be transformed in various ways and properties which hold through such deformations are an invariants.Idea 2: Semantic structures at the sentence level-for a typology of syntax structures.Two knots are equivalent when one curve can be deformed into the same shape as the other.Applications:Semantic transformations: invariant and variant (structures)Identic structures and interpretations in varying pragmatic contexts. 28
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In geometry: representational granularity.Verbal text are different representational types – different inferences.Diagram /depending on coordinatisation, surrounding topological spaces.Math FormulaMorning Star have same reference (Venus), but differ greatly in (mode of representation)Evening StarThe “2” has a sense (natural number which is the successor of “one”) and has reference (the number two).Энд ширээ байна.There is a table. Брат приехалЗдесь стол стoит. Брат приежалRepresentational similaritySpace-time dimensions of verbal (visual) perception. There are differences which are reflected in syntax structures, in grammatical categorization.Invariant of surrounding topological spaces intrinsic features of objects invariant in linguistics. 29
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Graph – as an ordered tripleIdea: Vertice point as a point for a coherence (logical) and cohesion (in three dimensional space).Next graph is a picture of the same graph as G in spite of their very differentappearances. Vertex – as a point where straight lines meet – lines of semantic force. to analyze a coherence. Proposition as a vector dimension, magnitude. Idea: to apply to discourse analysis, (comparison of semantic structures. (?) Knot theory. 30
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A weighted graph•is a graph for which each edge has an associated weight, usually given bya weight function w: E /discourse – set of proposition/Idea: weight of proposition –in Vector model-magnitude (size) ofsemantic force. (Measure of the length of a route, capacity of a line, the energy required to move between locations along a route). 31
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Isomorphism Problem Determining whether two graphs are isomorphic Although these graphs look very different, they are isomorphic; one isomorphism between them isComments: a h 1 2 g b 5 6 etc i 4 c 8Idea: Isomorphism between semantic structures of discourse – typology of discourse 32
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In quantum mechanics: Hilbert spaces are linear, vector spaces have infinite dimension = mental spaces have infinite dimensions. States (state of an electron) are represented by a vector Ѱ in the Hilbert space and properties (position, momentum, spin or energy of the electron) – by linear operators.Idea: Energy transmission (brain to brain) →mental spaces through which energy is transmitted should be modeled as a Hilbert spaces or vector spaces of infinite dimension.Structure component as a fermions (quarks, leptons)Semantic component as a bozon (force carrier particles, guaze bosons, photons, gluons, W and Z bosons)Photons – carriers of electromagnetic fieldW1Z – carriers of weak forceGluons – carriers of strong force 33
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Any prime of the form “4n+1” can be expressed as the sum of two perfect squares in one and only one way.13=4(3) + 1 = 9 + 4 = 32 + 22Idea: Rules for semantic transformation.To extend Chomsky UG - Rules for mental transformations (not logic rules, but some ideas of logic be applied) based on rules (mechanism) of perceptual spaces (modalities) – to serve as a basis for semantic transformations. 34
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Bolzano-Weierstrass theorem: Every bounded infinite set S has at least one cluster point. Q0 is divided into quarters. Q1 – into four quarters. Infinite sequence of subsets: … Q3 Q2 Q1 Q0. Each has many points of S. There is at least one point “P”, common to them all.Idea: Discourse structure → super proposition→ quarters as a part of discourse (in standard discourse) But: quarter is ?→ In case of non-standard discourse, where coordinatization systems aredifferent or Hilbert spaces of infinite dimensions. 35
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Perfect number is one which is equal to the sum of its positive divisors. This “6” is perfect number since it is divisible without remainder by 1, by 2, and by 3, and 1+2+3=6. Today about 45 (perfect number) are known. 21(22 - 1) = 6 to find some basic perfect numbers Perfect 22(23 - 1) = 28 (structures) at morphological, syntax number and discourse levels-structure 24(25 – 1) = 496 primitives to UG.Idea: deep structures are limited, mental structures are not limited.Idea: Math transformations should be applied to semantic (mental) transformations - New model for structural analysis. 36
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(-1, 1) (1, 1) Small sphere (-1, -1) (1, -1)Figure: with the small central sphere stay contained in the box?Distance from the origin to the centre of any sphere isEach sphere has radius 1; this radius of the central sphere is , for , there isIdea: Visual (verbal) interpretation and non-Euclidean (Elementary application tospace propositions, space expressions) geometry. (PoM. p 204) 37
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AoA hypothesis (age-of-acquisition) Synaptic plasticity → (synaptic pruning, distribution of neurotransmitter receptors, maturation of inhibition)Age of acquisition Cumulative frequency → (links between codes that are formedof words by the network become entrenched as a result of early experience. Later experience in the word frequencies do little to change themThe process of learning creates neurobiological changes that reduce plasticity. This is standard view applied to learning of language (not mathematics) – When to start SL (FL)Idea: Acquisition of word.Zero and first order tensors should serve modeling an acquisition of words at different stages of cognitive plasticity. 38
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Semantic relationships among words in Human mental space as a basis to form syntactic structures – be modeled as a vector having magnitude and direction.Scalar - words in mental space having only magnitude (size) – be modeled as a component of mental vocabulary. But there is only isolated component, no mention about a direction.Idea: Human mental vocabulary should be presented in vector and scalar models.Zero order tensor → Scalars weight (as a mass), electric charge (+ ; -)I order tensor → Vectors direction of cohesion. + р 11,15,16,29 39
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The inferior frontal gyrus (frontal operculum) was activated more strongly for semantically unrelated words and for words created syntactic violations – this activation was mostly bilateral, but stronger in left-hemisphere.Hypothesis: Shared syntactic integration-combination of discrete structural elements (words or musical tones) into sequences-perceiving complex acoustic, non-verbal structures (symbol?).New hypothesis: lack of musical priming in patients with Broca’s aphasia who have difficulties in Linguistic syntax processing. Холбох р 6,11,12,20,26Idea: acoustic structures –wave +particle.Verbal structures wave-particle (listening) particle (reading)X? carrier of semantic force (quant ?) 40
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- Arithmetic: musical processing-numerical processing Violation in number series S in sequential regularities-aphasia Violation in syntactic structures Violation in musical structures Musical structures (tone) Metrical stimuli: Syntactic structures (morphemes etc).Musical processing – perceiving waves (quantification of musical structures) + tonesas a particles.Dynamic attention – Cognitive sequencing.Idea: Experiment on universal syntax for verbal and musical production. Холбох р 25 41
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Inferential relations between: Sample-population Example-prototype Distance between “these…” Member-classInstance between member 1-class Member N-classIdea: Inferential relations and distance between invariant and variants (prototype – example etc) in Hilbert spaces (finite dimensional Hilbert space) + Bolzano’s theoremIdea: Inductive inference – isomorphism, homomorphism Холбох р 24(least square–for measuring a dispersion caused by an inference). 42
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Cognitive maps are mental representations. VisualPerception of spatial information VerbalSemantic relations between cities in case when spatial information (about locations – distance between cities) are presented:Idea: Symbol interdependency hypothesis - Language comprehension is both embodied and symbolic. Symbol-through interdependencies of a modal linguistic symbols. Embodied-through references these symbols make to perceptual representations Visual and verbal perception / recognition.In verbal for (discourse-text based)In visual form (geographic map-map based)Idea: - Projective geometry, representational granularity. -Vectors. 43
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Ability to identify stimuli: Types of stimuli (verbal, non-verbal etc). Features (simple, combined).Multidimensional stimuli – multidimensional scanning (MDS).Identification of multidimensional stimuli-performance limit 7± 2 (Miller) – MDS.Relationships between identification performance and structure (stimulus structure) StrongPsychological representations (simple and complex)Idea: Object recognition + projective geometry – representational granularity: set theory, graph theory. 44
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Conceptual metaphor (Cantor’s metaphor) “same number as is pairability”.Conceptual blending – BMI (the basic metaphor of infinity)Transfinite cardinals are the result of a combination of conceptual metaphor and conceptual blending (done by the creative mind of George Cantor).The underlying cognitive mechanisms are bodily-grounded and not arbitrary. This ground is constrained by biological phenomena such as neuroanatomy, the human nervous system.Source, input spaces, mappings, and projections are realized by bodily-grounded experience such as thermic experience, visual perception, spatial experience and so on.In the case of transfinite numbers these constraints are provided by container schemas.Idea: (be extended in difference versions) – human perceptual activity → verbal presentation and interpretation → mapping and blending in verbal forms (sentences etc) - associative mechanism for building a metaphor – is grounded (fox, head etc) 45
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Container schema: If A is in B and B is in C, then A is in C.Systems of mirror and canonical neurons points to joint action – perception circuitry – binding circuits - two primary metaphors form a complex metaphor – to interpret processes on the basis of container schema, inference by applying an algebra.Inference – аn operation serving metaphor building – embedding Life needs a container with anShould be modeled in terms of topological, interior, a boundary, and annon-Euclidean geometrical notions exterior. Container schema – is a spatial- relations concept, a gestalt + multimodality + embedding. 46
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Georg Cantor’s fundamental conceptual metaphor Same Number As is Pairability. This simple but ingenious metaphor is at the core of transfinite numbers and modern set/theory. Source domain Target domain Mappings Numeration Set A and Set B can be put Set A and Set B have the into 1-1 correspondence same number of elements. Set A and Set B cant be put Set B is larger than Set A. It in 1-1 correspondence, and has more elements than Set Set A is a proper subset of B. A.Isomorphism (Set theory)HomomorphismIdea: Cantor’s continuum hypothesis p 2, 4 47
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